Seconds, please?

A few posts ago I showed how we could find the probability of a player in position n in a game with p players finishing in first place. Now we can try and make that more general: what is the probability of a player in position n in a game with p players finishing in mth place?

Before we generalize it that far, however, let's first think about it for the case of the player getting second place. Let's suppose we want to find the probability that the player who goes second (in position two) in a game with four players finishes in second. What conditions have to be met for this to happen? Clearly, if the player is getting second, someone else is getting first, so one of the things that must happen is that someone besides that player wins the game. Let's consider what happens if the player going first gets first place. If we already know the player going first is going to win, then the chance that the player going second finishes in second in a game with four people is the same as the probability that a player going first in a game with three moves finishes in first place. Thus, the player going first gets first place and the player going second gets second place is equal to:

(Probability that player going first wins in a game with 4 players) * (Probability that a player going first wins in a game with three players)

This is getting pretty wordy, so let's invent some notation. Let's call F(n,p,m) the probability that a player in position n in a game with p players finishes in m place. The above expression can then be condensed to:

F(1,4,1)*F(1,3,1)

(Remember that we already found an algorithm that gives the probability of a player in position n in a game with p players finishing in first.)

         This has been pretty dense so far, so I think I'll stop here and continue on in the next post.